Haar system as Schauder basis in Besov spaces: The limiting cases for 0 < p <= 1

TL;DR

This paper establishes when the Haar system forms a Schauder basis in certain Besov spaces for 0 < p <= 1, particularly in limiting cases, resolving open questions from previous research.

Contribution

It determines the Schauder basis property of the Haar system in Besov spaces at critical smoothness levels, including previously unresolved cases for 0 < p < 1.

Findings

Haar system is a Schauder basis in B_{p,q,1}^s for 0 < p < 1 if 0 < q <= p.

Resolved the open case for d=1, 0 < p < q, in previous work.

Extended results to Fourier-analytic Besov spaces at critical smoothness levels.

Abstract

We show that the d-dimensional Haar system H^d on the unit cube I^d is a Schauder basis in the classical Besov space B_{p,q,1}^s(I^d), 0<p<1, defined by first order differences in the limiting case s=d(1/p-1), if and only if 0<q\le p. For d=1 and p<q, this settles the only open case in our 1979 paper [4], where the Schauder basis property of H in B_{p,q,1}^s(I) for 0<p<1 was left undecided. We also consider the Schauder basis property of H^d for the standard Besov spaces B_{p,q}^s(I^d) defined by Fourier-analytic methods in the limiting cases s=d(1/p-1) and s=1, complementing results by Triebel [7].